Theorem eulerthlem1 12181

Description: Lemma for eulerth 12187. (Contributed by Mario Carneiro, 8-May-2015.)

Hypotheses

Ref	Expression
eulerthlem1.1	|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
eulerthlem1.2	|- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
eulerthlem1.3	|- T = ( 1 ... ( phi ` N ) )
eulerthlem1.4	|- ( ph -> F : T -1-1-onto-> S )
eulerthlem1.5	|- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )

Assertion

Ref	Expression
eulerthlem1	|- ( ph -> G : T --> S )

Distinct variable groups:   x, y, A   x, F, y   x, G, y   x, N, y   x, S   ph, x, y   x, T, y

Allowed substitution hint:   S( y)

Proof of Theorem eulerthlem1

Step	Hyp	Ref	Expression
1		eulerthlem1.1	. . . . . . 7 |- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
2	1	simp2d 1005	. . . . . 6 |- ( ph -> A e. ZZ )
3	2	adantr 274	. . . . 5 |- ( ( ph /\ x e. T ) -> A e. ZZ )
4		eulerthlem1.4	. . . . . . . . . 10 |- ( ph -> F : T -1-1-onto-> S )
5		f1of 5442	. . . . . . . . . 10 |- ( F : T -1-1-onto-> S -> F : T --> S )
6	4, 5	syl 14	. . . . . . . . 9 |- ( ph -> F : T --> S )
7	6	ffvelrnda 5631	. . . . . . . 8 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. S )
8		oveq1 5860	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y gcd N ) = ( ( F ` x ) gcd N ) )
9	8	eqeq1d 2179	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( y gcd N ) = 1 <-> ( ( F ` x ) gcd N ) = 1 ) )
10		eulerthlem1.2	. . . . . . . . 9 |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
11	9, 10	elrab2 2889	. . . . . . . 8 |- ( ( F ` x ) e. S <-> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
12	7, 11	sylib 121	. . . . . . 7 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
13	12	simpld 111	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ( 0..^ N ) )
14		elfzoelz 10103	. . . . . 6 |- ( ( F ` x ) e. ( 0..^ N ) -> ( F ` x ) e. ZZ )
15	13, 14	syl 14	. . . . 5 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ZZ )
16	3, 15	zmulcld 9340	. . . 4 |- ( ( ph /\ x e. T ) -> ( A x. ( F ` x ) ) e. ZZ )
17	1	simp1d 1004	. . . . 5 |- ( ph -> N e. NN )
18	17	adantr 274	. . . 4 |- ( ( ph /\ x e. T ) -> N e. NN )
19		zmodfzo 10303	. . . 4 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
20	16, 18, 19	syl2anc 409	. . 3 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
21		modgcd 11946	. . . . 5 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
22	16, 18, 21	syl2anc 409	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
23	17	nnzd 9333	. . . . . 6 |- ( ph -> N e. ZZ )
24	23	adantr 274	. . . . 5 |- ( ( ph /\ x e. T ) -> N e. ZZ )
25	16, 24	gcdcomd 11929	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) gcd N ) = ( N gcd ( A x. ( F ` x ) ) ) )
26	23, 2	gcdcomd 11929	. . . . . . 7 |- ( ph -> ( N gcd A ) = ( A gcd N ) )
27	1	simp3d 1006	. . . . . . 7 |- ( ph -> ( A gcd N ) = 1 )
28	26, 27	eqtrd 2203	. . . . . 6 |- ( ph -> ( N gcd A ) = 1 )
29	28	adantr 274	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd A ) = 1 )
30	24, 15	gcdcomd 11929	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = ( ( F ` x ) gcd N ) )
31	12	simprd 113	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) gcd N ) = 1 )
32	30, 31	eqtrd 2203	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = 1 )
33		rpmul 12052	. . . . . 6 |- ( ( N e. ZZ /\ A e. ZZ /\ ( F ` x ) e. ZZ ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
34	24, 3, 15, 33	syl3anc 1233	. . . . 5 |- ( ( ph /\ x e. T ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
35	29, 32, 34	mp2and 431	. . . 4 |- ( ( ph /\ x e. T ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 )
36	22, 25, 35	3eqtrd 2207	. . 3 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 )
37		oveq1 5860	. . . . 5 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( y gcd N ) = ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) )
38	37	eqeq1d 2179	. . . 4 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( ( y gcd N ) = 1 <-> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
39	38, 10	elrab2 2889	. . 3 |- ( ( ( A x. ( F ` x ) ) mod N ) e. S <-> ( ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) /\ ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
40	20, 36, 39	sylanbrc 415	. 2 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. S )
41		eulerthlem1.5	. 2 |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )
42	40, 41	fmptd 5650	1 |- ( ph -> G : T --> S )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   {crab 2452   |-> cmpt 4050   -->wf 5194   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   x. cmul 7779   NNcn 8878   ZZcz 9212   ...cfz 9965  ..^cfzo 10098   mod cmo 10278   gcd cgcd 11897   phicphi 12163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  eulerthlemh  12185  eulerthlemth  12186